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Properties

Label 1-671-671.112-r0-0-0
Degree $1$
Conductor $671$
Sign $0.998 + 0.0547i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.994 − 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 − 0.207i)4-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)6-s + (0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (0.406 − 0.913i)10-s + (0.913 + 0.406i)12-s + (−0.913 + 0.406i)13-s + (0.978 − 0.207i)14-s + (0.913 − 0.406i)15-s + (0.913 − 0.406i)16-s + (−0.866 + 0.5i)17-s + (0.406 + 0.913i)18-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 − 0.207i)4-s + (0.5 − 0.866i)5-s + (0.866 + 0.5i)6-s + (0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (0.406 − 0.913i)10-s + (0.913 + 0.406i)12-s + (−0.913 + 0.406i)13-s + (0.978 − 0.207i)14-s + (0.913 − 0.406i)15-s + (0.913 − 0.406i)16-s + (−0.866 + 0.5i)17-s + (0.406 + 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.998 + 0.0547i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.998 + 0.0547i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.075343319 + 0.1117132402i\)
\(L(\frac12)\) \(\approx\) \(4.075343319 + 0.1117132402i\)
\(L(1)\) \(\approx\) \(2.719470290 + 0.04461498505i\)
\(L(1)\) \(\approx\) \(2.719470290 + 0.04461498505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.994 - 0.104i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (-0.743 - 0.669i)T \)
31 \( 1 + (-0.207 - 0.978i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.994 + 0.104i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (-0.207 - 0.978i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.7067676628688626710855592667, −21.86218539362698348298993272796, −21.34176932740744695501657514760, −20.23379384402694553789216151546, −19.8738000186296952925552547129, −18.64718999556889867220365330882, −17.8897221352264292459583231689, −17.175036399498154720306142409706, −15.68768537752729717736145434094, −14.936940037313115583038875663317, −14.43162442926784195353905504380, −13.770323415266320527677926719152, −12.98172831359199413929640466562, −12.04541779345201394885718627663, −11.16999208083837746137320177221, −10.30474024843038114564436203249, −9.0432885741987523348013348845, −7.96042424004063777970446609630, −7.12305448279877772212867531669, −6.5729057050004014121813530385, −5.32817176231155492958311638985, −4.42828077045875679155787247932, −3.11634151622303603840127510874, −2.44922313915381202265393808485, −1.6728471728929197722042253861, 1.8696790766295002058921641122, 2.05007860639502048899707632697, 3.692794207791983719003882255284, 4.42193641254685567940390438112, 5.103636720652520151003797090704, 6.027295247802007598112572340168, 7.522166964238902612688704901045, 8.156938304453811088136569710021, 9.32216016699380582705830912741, 10.09280163622605361324472881956, 11.109062324564139437558253627486, 12.005510622112500972487809403286, 13.01904802812881736797888789065, 13.68058969907740252786015519549, 14.50083356210586327381460823841, 15.011541980782554982181261761018, 16.035369038069608831222033493792, 16.78352126050600711322885961876, 17.61481273259399056882988028713, 19.09631411239832397518239598454, 19.91114036830820684394636472496, 20.533303616186964822454650489120, 21.22977249539442074953833535921, 21.66471511687810495394770629338, 22.54844300088799579993392963252

Graph of the $Z$-function along the critical line